There are 9,979,200
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There are 11 factorial, or 39,916,800 permutations of the letters in the word PROBABILITY. However, because the letters B and I are duplicated, there are only one quarter of that, or 9,979,200 distinguishable permutations.
The number of permutations of the letters in the word LOUISIANA is 9 factorial or 362,880. However, since the letters I and A are each repeated once, you need to divide that by 4 to determine the number of distinct permutations, giving you 90,720.
The number of permutations of the letters SWIMMING is 8 factorial or 40,320. The number of distinct permutations, however, due to the duplication of the letters I and M is a factor of 4 less than that, or 10,080.
There are 5*4*3 = 60 permutations.